Cauchy Solution of the Kinematic Wave Shallow-Water Equations Using Square-Grid Finite-Element Method

Abstract

AbstractThe kinematic wave system of shallow-water equations is posed as an initial value problem, or Cauchy problem, to investigate error analysis using the square-grid finite-element method. The consistent, lumped, and upwind models of the square finite-element method are analyzed using the Fourier (or von Neumann) stability method as separate initial value problems for two-dimensional shallow-water equations. The small difference between the Cauchy solution and integer (or half) multiples of the eigenvalue solution of amplification factors using the eigenvalue method indicates that the Cauchy solution is a possible stable numerical solution of the kinematic wave shallow-water equations within the reasonable limits of solution accuracy. The nodal amplification factors are less than or equal to unity for implicit finite-element schemes for all of the three formulations—consistent, lumped, and upwind for all of the wave numbers, implying unconditional stability. For explicit and semi-implicit finite-eleme...